Fig. REF  illustrates the robustness of the proposed scheme to channel estimation error. The sum rate is depicted versus the number of IRS elements (\(M\) ) for three cases of robust design, non-robust design and perfect CSI. In the non-robust design, it is assumed that the value of \(\sigma ^2=0\) , hence we can deploy this scenario by considering the value of \(\sigma _{h}^2\)  in the design process to be zero, while channel estimation error exists, i. e., \(\sigma _{AU}^2\)  and \(\sigma _{BU}^2\)  are non-zero, and it effects the output rate.
It is shown that the performance of the proposed robust method is better than the non-robust case. Note that, as \(M\)  increases, the gap between the robust design and the non-robust structure widens, which is a result of the approximation in (REF ), since the approximation in (REF ) is accurate when \(M\)  is large [1]}. Moreover, as the channel estimation error increases, the performance gain of the robust design over non-robust system becomes more obvious.
